Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2014 / 3 / a / i / Solution

The rank-nullity theorem states that for a linear map $A:X\to Y$ with finite-dimensional domain,Equivalently, nullity plus rank equals the dimension of the domain. One proof extends a basis of $\ker A$ to a basis of $X$; the images of the added vectors form a basis of $\mathrm{im}A$.